Algebra of physical space

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In physics, the algebra of physical space (APS) is the Clifford algebra (Geometric algebra) Cl3 of the three-dimensional Euclidean space with emphasis in its paravector structure.

The most compact faithful presentation of the Cl3 algebra can be obtained with the Pauli matrices.

The virtue of APS is that it can be used to construct a highly compact and unified formalism for both classical and quantum mechanics with an additional striking geometrical interpretation.

APS should not be confused with the space-time algebra (STA), initially developed and promoted by David Hestenes.

Contents

[edit] Special Relativity

In APS, the space-time position is represented as a paravector with the following matrix representation in terms of the Pauli matrices

x = \begin{pmatrix} t + z && x - iy \\ x + iy && t-z \end{pmatrix}

The four-velocity also called proper velocity is represented by a unimodular paravector u\, that transforms under the action of the Lorentz rotor L as

u \rightarrow u^\prime = L u L^\dagger.

The Lorentz rotor is chosen to be isomorphic to the SL(2C) group, which is the double cover of the Lorentz-group. If the transformation only involves space rotations, the Lorentz rotor belongs to the smaller compact group SU(2).

[edit] Classical Electrodynamics

The electromagnetic field is represented as a bi-paravector F. The Maxwell equations can be expressed in a single equation as follows

\bar{\partial} F = \frac{1}{c \epsilon} \bar{j},

where the overbar represents the Clifford conjugation.

The Lorentz force equation takes the form

\frac{d p}{d \tau} = \langle F u \rangle_{\Re}

[edit] Relativistic Quantum Mechanics

The Dirac equation takes the form

i \bar{\partial} \Psi\mathbf{e}_3 + e \bar{A} \Psi = m \bar{\Psi}^\dagger ,

where \mathbf{e}_3 is an arbitrary unitary vector and A is the paravector potential that includes the vector potential and the electric potential.

[edit] Classical Spinor

The differential equation of the Lorentz rotor that is consistent with the Lorentz force is

\frac{d \Lambda}{ d \tau} = \frac{e}{2mc} F \Lambda,

such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest

u = \Lambda \Lambda^\dagger,

which can be integrated to find the space-time trajectory.

[edit] See also

[edit] References

[edit] Textbooks

  • Baylis, William (2002). Electrodynamics: A Modern Geometric Approach (2th ed.). Birkhäuser. ISBN 0-8176-4025-8
  • W. E. Baylis, editor, Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering, Birkhäuser, Boston 1996.
  • Chris Doran and Anthony Lasenby, Geometric Algebra for Physicists, Cambridge University Press (2003)
  • David Hestenes: New Foundations for Classical Mechanics (Second Edition). ISBN 0-7923-5514-8, Kluwer Academic Publishers (1999)

[edit] Articles

  • Baylis, William (2002). Relativity in Introductory Physics, Can. J. Phys. 82 (11), 853--873 (2004). (ArXiv:physics/0406158)
  • W. E. Baylis and G. Jones, The Pauli-Algebra Approach to Special Relativity, J. Phys. A22, 1-16 (1989)
  • W. E. Baylis, Classical eigenspinors and the Dirac equation ,Phys Rev. A, Vol 45, number 7 (1992)
  • W. E. Baylis, Relativistic dynamics of charges in electromagnetic fields: An eigenspinor approach ,Phys Rev. A, Vol 60, number 2 (1999)